Quantization Of Lie Bialgebras Research Articles

Quantization of the Rank Two Heisenberg–Virasoro Algebra

Quantum groups occupy a significant position in both mathematics and physics, contributing to progress in these fields. It is interesting to obtain new quantum groups by the quantization of Lie bialgebras. In this paper, the quantization of the rank two Heisenberg–Virasoro algebra by Drinfel’d twists is presented, Lie bialgebra structures of which have been investigated by the authors recently.

Over then under tangles

Over-then-Under (OU) tangles are oriented tangles whose strands travel through all of their over crossings before any under crossings. In this paper, we discuss the idea of gliding: an algorithm by which tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain a braid classification result, which we also extend to virtual braids, and provide a Mathematica implementation. We discuss other instances of successful “gliding ideas” in the literature — sometimes in disguise — such as the Drinfel’d double construction, Enriquez’s work on quantization of Lie bialgebras, and Audoux and Meilhan’s classification of welded homotopy links.

Quantizations of Lie bialgebras, duality involution and oriented graph complexes

We prove that the action of the Grothendieck–Teichmüller group on the genus completed properad of (homotopy) Lie bialgebras commutes with the reversing directions involution of the latter. We also prove that every universal quantization of Lie bialgebras is homotopy equivalent to the one which commutes with the duality involution exchanging Lie bracket and Lie cobracket. The proofs are based on a new result in the theory of oriented graph complexes (which can be of independent interest) saying that the involution on an oriented graph complex that changes all directions on edges induces the identity map on its cohomology.

Classification of universal formality maps for quantizations of Lie bialgebras

We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor ${\mathcal {D}}$ in the category of augmented props with the property that for any representation of a prop ${\mathcal {P}}$ in a vector space $V$ the associated prop ${\mathcal {D}}{\mathcal {P}}$ admits an induced representation on the graded commutative algebra $\odot ^\bullet V$ given in terms of polydifferential operators. Applying this functor to the minimal resolution $\widehat {\mathcal {L}\textit{ieb}}_\infty$ of the genus completed prop $\widehat {\mathcal {L}\textit{ieb}}$ of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props \[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \] satisfying certain boundary conditions, where $\mathcal {A}\textit{ssb}_\infty$ is a minimal resolution of the prop of associative bialgebras. We prove that the set of such formality morphisms is non-empty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator $\mathfrak{A}$ there is an associated ${\mathcal {L}} ie_\infty$ quasi-isomorphism between the ${\mathcal {L}} ie_\infty$ algebras $\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$ and $\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$ controlling, respectively, deformations of the standard bialgebra structure in $\odot V$ and deformations of any given Lie bialgebra structure in $V$. We study the deformation complex of an arbitrary universal formality morphism $\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$ and prove that it is quasi-isomorphic to the full (i.e. not necessary connected) version of the graph complex introduced Maxim Kontsevich in the context of the theory of deformation quantizations of Poisson manifolds. This result gives a complete classification of the set $\{F_\mathfrak{A}\}$ of gauge equivalence classes of universal Lie connected formality maps: it is a torsor over the Grothendieck–Teichmüller group $GRT=GRT_1\rtimes {\mathbb {K}}^*$ and can hence can be identified with the set $\{\mathfrak{A}\}$ of Drinfeld associators.

Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial

This is the first in a series of papers studying w-knotted objects (w-knots, w-braids, w-tangles, etc.), which make a class of knotted objects which is {w}ider but {w}eaker than their usual counterparts. The group of w-braids was studied (as "{w}elded braids") by Fenn-Rimanyi-Rourke and was shown to be isomorphic to the McCool group of "basis-conjugating" automorphisms of a free group Fn. Brendle-Hatcher, tracing back to Goldsmith, have shown this group to be a group of movies of flying rings in R3. Satoh studied several classes of w-knotted objects (as "{w}eakly-virtual") and has shown them to be closely related to certain classes of knotted surfaces in R4. So w-knotted objects are algebraically and topologically interesting. Here we study finite type invariants of w-knotted objects. Following Berceanu-Papadima, we construct homomorphic universal finite type invariants ("expansions") of w-braids and of w-tangles. We find that the universal finite type invariant of w-knots is essentially the Alexander polynomial. We find that the spaces Aw of "arrow diagrams" for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic expansion of w-knotted foams is essentially the same as a solution of the Kashiwara-Vergne conjecture (KV), thus giving a topological explanation to the work of Alekseev-Torossian work on KV and Drinfel'd associators. The true value of w-knots, though, is likely to emerge later, for we expect them to serve as a {w}armup example for the study of virtual knots. We expect v-knotted objects to provide the global context whose associated graded structure will be the Etingof-Kazhdan theory of quantization of Lie bialgebras.

Quantization of Lie bialgebras revisited

We describe a new method of quantization of Lie bialgebras, based on a construction of Hopf algebras out of a cocommutative coalgebra and a braided comonoidal functor.

The Oriented Graph Complexes

Oriented graph complexes, in which graphs are not allowed to have oriented cycles, govern for example the quantization of Lie bialgebras and infinite dimensional deformation quantization. It is shown that the oriented graph complex GC^or_n is quasi-isomorphic to the ordinary commutative graph complex GC_{n-1}, up to some known classes. This yields in particular a combinatorial description of the action of the Grothendieck-Teichm\"uller Lie algebra on Lie bialgebras, and shows that a cycle-free formality morphism in the sense of Shoikhet can be constructed rationally without reference to configuration space integrals.

CLASSIFICATION OF FLAT VIRTUAL PURE TANGLES

Virtual knot theory, introduced by Kauffman [Virtual Knot theory, European J. Combin.20 (1999) 663–690, arXiv:math.GT/9811028], is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation [D. Bar-Natan, u, v, w-knots: Topology, Combinatorics and low and high algebra] of Etingof and Kazhdan's theory of quantization of Lie bi-algebras [Quantization of Lie Bialgebras, I, Selecta Math. (N.S.) 2 (1996) 1–41, arXiv:q-alg/9506005]. Classical knots inject into virtual knots [G. Kuperberg, What is Virtual Link? Algebr. Geom. Topol.3 (2003) 587–591, arXiv:math.GT/0208039], and flat virtual knots [V. O. Manturov, On free knots, preprint (2009), arXiv:0901.2214; On free knots and links, preprint (2009), arXiv:0902.0127] is the quotient of virtual knots which equates the real positive and negative crossings, and in this sense is complementary to classical knot theory within virtual knot theory. We completely classify flat virtual tangles with no closed components (pure tangles). This classification can be used as an invariant on virtual pure tangles and virtual braids.

Quantization of Lie Bialgebras, Part VI: Quantization of Generalized Kac–Moody Algebras

This paper is a continuation of the series of papers “Quantization of Lie bialgebras (QLB) I-V”. We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with the tensor category structure defined by a Drinfeld associator. This equivalence is a generalization of the functor constructed previously by G. Lusztig and the second author. In particular, we answer positively a question of Drinfeld whether the characters of irreducible highest weight modules for quantized Kac-Moody algebras are the same as in the classical case. Moreover, our results are valid for the Lie algebra \(\mathfrak{g}(A)\) corresponding to any symmetrizable matrix A (not necessarily with integer entries), which answers another question of Drinfeld. We also prove the Drinfeld-Kohno theorem for the algebra \(\mathfrak{g}(A)\) (it was previously proved by Varchenko using integral formulas for solutions of the KZ equations).

Invariant star products on nondegenerate triangular Lie bialgebras over formal power series rings

We obtain a bijection between the set of equivalent classes of invariant star products on a non-degenerate triangular finite dimensional Lie bialgebra over the formal power series ring and the set , working in the framework developed by Etingof–Kazhdan for the quantization of Lie bialgebras. Two of the corresponding triangular Hopf algebras over the ring are isomorphic if and only if the invariant star products defining them are equivalent. Therefore, when t = ℏ, we obtain a set of triangular Hopf quantized universal enveloping algebras which can also be seen as quantizations of the deformation algebra . Additionally, two of them are isomorphic if and only if the above invariant star products are equivalent.

Quantization of lie Bialgebras via the Formality of the operad of Little Disks

We give a proof of the Etingof–Kazhdan theorem on quantization of Lie bialgebras based on the formality of the chain operad of little disks and show that the Grothendieck–Teichmuller group acts non-trivially on the corresponding quantization functors.

On the invertibility of quantization functors

Certain quantization problems are equivalent to the construction of morphisms from “quantum” to “classical” props. Once such a morphism is constructed, Hensel's lemma shows that it is in fact an isomorphism. This gives a new, simple proof that any Etingof–Kazhdan quantization functor is an equivalence of categories between quantized universal enveloping (QUE) algebras and Lie bialgebras over a formal series ring (dequantization). We apply the same argument to construct dequantizations of formal solutions of the quantum Yang–Baxter equation and of quasitriangular QUE algebras. We derive from there a classification of all twistors killing a given associator. We also give structure results for the props involved in quantization of Lie bialgebras, which yield an associator-independent proof that the prop of QUE algebras is a flat deformation of the prop of co-Poisson universal enveloping algebras.

Double product integrals and Enriquez quantization of Lie bialgebras I: The quasitriangular identities

Let 𝒯(ℒ) be the space of all tensors over a Lie algebra ℒ in which the Lie bracket is obtained by taking commutators in an associative algebra. We show that 𝒯(ℒ) becomes a Hopf algebra when equipped with a noncommutative modification of the shuffle product together with the standard coproduct. A definition is given of directed double product integrals as iterated single product integrals driven by formal power series with coefficients in the tensor product of ℒ with an appropriate associative algebra. For the Hopf algebra T(L)[[h]] of formal power series we show that elements R[h] of (T(L)⊗T(L))[[h]] satisfying (Δ⊗id)R[h]=R[h]13R[h]23, (id⊗Δ)R[h]=R[h]13R[h]12,and which are unitalized by the counit in either copy of 𝒯(ℒ), can be characterized as such directed double product integrals ∏∏(1+d⃗⊗d↙r[h]) where r[h] is a formal power series with coefficients in L⊗L and vanishing constant term.

Comment on “Double product integrals and Enriquez quantization of Lie bialgebras I: The quasitriangular identities” [Hudson and Pulmannová, J. Math. Phys. 45, 2090 (2004)

Comment on “Double product integrals and Enriquez quantization of Lie bialgebras I: The quasitriangular identities” [Hudson and Pulmannová, J. Math. Phys. <b>45</b>, 2090 (2004)

Quantization of Lie bialgebras and shuffle algebras of Lie algebras

To any field \( \Bbb K \) of characteristic zero, we associate a set \( (\mathbb{K}) \) and a group \( {\cal G}_0(\Bbb K) \). Elements of \( (\mathbb{K}) \) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of \( {\cal G}_0(\Bbb K) \) are universal automorphisms of the adjoint representations of Lie bialgebras over \( \Bbb K \). We construct a bijection between \( (\mathbb{K})\times{\cal G}_0(\Bbb K) \) and the set of quantization functors of Lie bialgebras over \( \Bbb K \). This construction involves the following steps.¶ 1) To each element \( \varpi \) of \( (\mathbb{K}) \), we associate a functor \( \frak a\mapsto\operatorname{Sh}^\varpi(\frak a) \) from the category of Lie algebras to that of Hopf algebras; \( \operatorname{Sh}^\varpi(\frak a) \) contains \( U\frak a \).¶ 2) When \( \frak a \) and \( \frak b \) are Lie algebras, and \( r_{\frak a\frak b} \in\frak a\otimes\frak b \), we construct an element \( {\cal R}^{\varpi} (r_{\frak a\frak b}) \) of \( \operatorname{Sh}^\varpi(\frak a)\otimes\operatorname{Sh}^\varpi(\frak b) \) satisfying quasitriangularity identities; in particular, \( {\cal R}^\varpi(r_{\frak a\frak b}) \) defines a Hopf algebra morphism from \( \operatorname{Sh}^\varpi(\frak a)^* \) to \( \operatorname{Sh}^\varpi(\frak b) \).¶ 3) When \( \frak a = \frak b \) and \( r_\frak a\in\frak a\otimes\frak a \) is a solution of CYBE, we construct a series \( \rho^\varpi(r_\frak a) \) such that \( {\cal R}^\varpi(\rho^\varpi(r_\frak a)) \) is a solution of QYBE. The expression of \( \rho^\varpi(r_\frak a) \) in terms of \( r_\frak a \) involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing statements for cohomologies arising from universal algebras for the solutions of CYBE.¶ 4) We define the quantization of a Lie bialgebra \( \frak g \) as the image of the morphism defined by \( {\cal R}^\varpi(\rho^\varpi(r)) \), where \( r \in \mathfrak{g} \otimes \mathfrak{g}^* \).

Quantization of Lie bialgebras, Part IV: The coinvariant construction and the quantum KZ equations

Quantization of Lie bialgebras, Part IV: The coinvariant construction and the quantum KZ equations

Quantization of Lie bialgebras, Part V: Quantum vertex operator algebras

Quantization of Lie bialgebras, Part V: Quantum vertex operator algebras

Quantization of Lie bialgebras, III

Quantization of Lie bialgebras, III

Quantization of Lie bialgebras, II

In the paper [Dr3] V. Drinfeld formulated a number of problems in quantum group theory. In particular, he raised the question about the existence of a quantization for Lie bialgebras, which arose from the problem of quantization of Poisson Lie groups. When the paper [KL] appeared Drinfeld asked whether the methods of [KL] could be useful for the problem of quantization of Lie bialgebras. This paper gives a positive answer to a number of Drinfeld's questions, using the methods and ideas of [KL]. In particular, we show the existence of a quantization for Lie bialgebras. The universality and functoriality properties of this quantization will be discussed in the second paper of this series. We plan to provide positive answers to most of the remaining questions in [Dr3] in the following papers of this series.

Quantization of Lie bialgebras, I

In the paper [Dr3] V. Drinfeld formulated a number of problems in quantum group theory. In particular, he raised the question about the existence of a quantization for Lie bialgebras, which arose from the problem of quantization of Poisson Lie groups. When the paper [KL] appeared Drinfeld asked whether the methods of [KL] could be useful for the problem of quantization of Lie bialgebras. This paper gives a positive answer to a number of Drinfeld's questions, using the methods and ideas of [KL]. In particular, we show the existence of a quantization for Lie bialgebras. The universality and functoriality properties of this quantization will be discussed in the second paper of this series. We plan to provide positive answers to most of the remaining questions in [Dr3] in the following papers of this series.